US2004/0168664A1 (JP-A-2004-257369) discloses an auxiliary system that includes an engine starter supplying electricity to a starter motor for starting an engine.
As shown in FIG. 7 a conventional engine-starting device includes switches 101, 102, a starter relay 103, a starter 104, and an engine-ECU 105.
The switch 101 is turned ON when a key cylinder is rotated to a start position. The switch 102 is turned ON, when a gear is in a neutral position or when the gear is in a parking position in an automatic-transmission car. Alternatively, the switch 102 is turned ON when a clutch is stepped in a manual-transmission car. The starter relay 103 includes a winding 103a and a relay switch 103b. The starter 104 includes a starter motor 104a and a starter coil 104b. 
In this structure, when the gear is in the neutral position or in the parking position in the automatic-transmission car, the switch 102 is turned ON. Alternatively, when the clutch is stepped in the manual-transmission car, the switch 102 is turned ON. In these conditions, when the driver rotates the key cylinder to the start position, a signal is transmitted to the engine-ECU 105, so that the engine-ECU 105 supplies electricity from a power source 106 via a terminal 108, and the engine-ECU 105 outputs a signal for fuel injection in the engine.
Thus, the winding 103a of the starter relay 103 is supplied with electricity via the switch 102, so that the winding 103a generates electromagnetic force that attracts the starter switch 103b. The power source 106 supplies electricity to the starter coil 104b and the starter motor 104a of the starter 104, so that the engine is started. The engine-ECU 105 supplies electricity in accordance with a condition of the power source (battery). When the engine is started, voltage (battery voltage) of the power source 106 quickly decreases. As shown in FIG. 5, when the battery voltage is less than a predetermined reset level (engine-ECU reset level) at the timing T1, the battery voltage is insufficient for operating the engine-ECU 105. Accordingly, the engine-ECU 105 resets electricity supply, so that the engine-ECU 105 terminates supplying electricity to the winding 103a. 
When the key cylinder is rotated to the start position, the switch 101 is turned ON, so that a terminal 107 supplies electricity from the power source 106, and the starter motor 104a is rotated. Thus, an engine starting load is gradually reduced while the starter motor 104a is rotated, so that the battery voltage gradually recovers in the period between T1 and T2 in FIG. 5. When the battery voltage increases to a recovery level (engine-ECU recovery level), the reset condition of the engine-ECU 105 is cancelled at the timing T2, subsequently the engine is restarted at the timing T3.
Thus, the conventional engine-starting device can start the engine, even in the case where the battery voltage decreases in engine starting.
However, an engine-starting device may have a structure, in which the terminal 107 cannot supply electricity from the power source 106 when the key cylinder is turned to the ON position.
In recent years, an engine-starting device including a push switch to start an engine is developed. This engine-starting device does not have a path such as a mechanical contact, through which the terminal 107 supplies electricity from the power source 106 to the starter relay 103, when the engine-ECU 105 resets electricity supply. Accordingly, this engine-starting device cannot start the engine when the engine-ECU 105 resets electricity supply.
In this engine-starting device, when the engine-ECU 105 resets electricity supply and engine starting is aborted, the battery voltage recovers to a level before the engine is started. However, when the engine is started again, the battery voltage decreases to be less than the reset level, because a load for staring engine is not reduced. Accordingly, even when the push switch is pushed over and over again, recovery and decrease in the battery voltage are repeated. As a result, the engine cannot be started.